The London Stock Exchange has rejected a £1.35bn takeover approach from German rival Deutsche Borse.
The LSE said the 530p-a-share proposal undervalued both the company and potential savings from a tie-up with the Frankfurt-based exchange.
Concerns that a takeover would face competition and regulatory issues also led LSE directors to reject the approach.
But the group said in a statement that it had agreed to hold talks with Deutsche Borse to see whether it could return with a significantly improved proposal.
The LSE has long been seen as a target for Deutsche Borse, which has a market value of about £3.13bn, and the two companies were poised to merge four years ago.
At that time, the LSE was owned by its members, who decided that a deal would not be in their interests.
The deal also foundered in the face of a hostile bid for the LSE from Stockholm Stock Exchange operator OM Group.
Competition concerns arise from Europe having only three major exchanges - the LSE, Deutsche Borse and pan-European operation Euronext.
A formal offer from Deutsche Borse could spark a bidding war if Euronext decided to join the fight.
The LSE is thought to fear a drawn-out wrangle with the European Commission if it recommends an offer from Deutsche Borse, although it concluded four years ago that a merger would be approved.
Shares in the LSE have risen 25 per cent in recent weeks as investors anticipated a possible takeover bid and are likely to climb even higher today.
In a statement, LSE directors said they believed in the company's strong growth prospects as the largest equity market in Europe with a unique position in global capital markets.
They said: "The board of LSE is fully committed to continuing to deliver value to shareholders and customers as an independent group."
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